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or superscriptǫ. Note that slow manifolds are defined by asymptotic conditions like (F2). Furthermore, their non-uniqueness (F6) and finite smoothness (F4) are not sur- prising as the proof constructs them as center-type manifolds intersecting the attracting and repelling manifolds of the fast subsystem; cf. [16, 42, 103]. In addition to critical and slow manifolds we can also consider their associated attracting and repelling manifolds. If S 0 is normally hyperbolic we view points on it as hyperbolic equilibria of the fast subsystem and define W s (S 0)= W s (p), W u (S 0)= W u (p) p∈S 0 Sometimes we refer to W s (S 0) and W u (S 0) as stable and unstable manifolds to the critical manifold. Theorem 1.1.2 (Fenichel’s Theorem, Part 2). Forǫ> 0 sufficiently small there exist manifolds W s (Sǫ) and W u (Sǫ). The conclusions (F1)-(F6) hold for W s (Sǫ) and W u (Sǫ) if we replace Sǫ and S 0 by W s (Sǫ) and W s (S 0) (or similarly by W u (Sǫ) and W u (S 0)). In addition to Fenichel’s Theorem we can also find coordinate changes that simplify a fast-slow system considerably near a critical manifold. Theorem 1.1.3. (Fenichel Normal Form, [42, 72]) Suppose the origin 0∈Cis a normally hyperbolic point with ms stable and mu unstable fast directions. Then there exists a smooth invertible coordinate change (x, y)↦→ (a, b, v)∈R ms+mu+n so that a fast-slow system (6.2) can be written as: a ′ = Λ(a, b, v,ǫ)a p∈S 0 b ′ = Γ(a, b, v,ǫ)b (1.9) v ′ = ǫ(m(v,ǫ)+ H(a, b, v,ǫ)ab) 7
whereΛ,Γare matrix-valued functions and H is bilinear and given in coordinates by 1.1.3 Canards ms mu Hi(a, b, v,ǫ)ab= r=1 s=1 Hirsarbs (1.10) We shall now consider different types of trajectories in Van der Pol’s equation. Consider first the caseλ = 0 (“unforced Van der Pol equation”). Forǫ = 0 the fast and slow flows are indicated in Figure 1.1. A singular orbit can be constructed by concatenating fast and slow flow segments as shown in Figure 1.1(a); note that singular orbits are also called candidates. The singular orbit follows the slow flow on C VdP , then reaches a fold point, “jumps” and follows the fast subsystem until it reaches another branch of the critical manifold. The same mechanism returns the orbit to the initial branch of the critical manifold. It can be shown [93, 86] that the singular orbit perturbs forǫ> 0 and we have the classical scenario of relaxation oscillations. Note that (x, y)=(0, 0) is an unstable focus forλ=0. A direct linear stability analysis shows that forλ>1 the unique equilibrium point (x, y)=(λ,λ 3 /3−λ)=: q is a stable focus and undergoes su- percritical Hopf bifurcation atλH= 1. The precise analysis of the orbit structure of Van der Pol’s equation between the stable focus region and relaxation oscillations was one of the first major steps in the theory of fast-slow systems. Figure 1.2(a) shows numerical continuation results forǫ = 0.05. The key feature is that the amplitude of the periodic orbits generated in the Hopf bifurcation grows rapidly in an exponentially small interval in parameter space. This process is called canard explosion and we refer to the Hopf bifurcation as 8
Page 1 and 2: MULTIPLE TIME SCALE DYNAMICS WITH T
Page 3 and 4: MULTIPLE TIME SCALE DYNAMICS WITH T
Page 5 and 6: To my family. iv
Page 7 and 8: ing mechanical systems.”, [E.T. P
Page 9 and 10: TABLE OF CONTENTS Biographical Sket
Page 11 and 12: LIST OF TABLES 5.1 Euclidean distan
Page 13 and 14: 3.1 Bifurcation diagram of (3.6). H
Page 15 and 16: 4.6 Boundary conditions are blue an
Page 17 and 18: 6.3 For all computations of the ful
Page 19 and 20: where (x, y)∈R m × R n are varia
Page 21 and 22: the general definition of normal hy
Page 23: fast-slow system and yields: ˙x=(D
Page 27 and 28: forǫ> 0. Therefore they have no im
Page 29 and 30: The Hopf bifurcation is supercritic
Page 31 and 32: where the algorithm can be used to
Page 33 and 34: slow flow. In particular the manifo
Page 35 and 36: f ast whereφ t is the flow of the
Page 37 and 38: whereµ∈R p are parameters. Usual
Page 39 and 40: We shall not prove this result but
Page 41 and 42: d as given in equation (2.5). Let p
Page 43 and 44: variables x1= u, x2= v and y=w we g
Page 45 and 46: The geometry of the system for one
Page 47 and 48: (0, 0, 0). 2.3 The Exchange Lemma I
Page 49 and 50: withδ>0 sufficiently small. The sa
Page 51 and 52: Theorem 2.3.2. Let ¯q ∈ M∩{|a|
Page 53 and 54: exit point of a trajectory starting
Page 55 and 56: Hence we have k=1=l and n=2 in view
Page 57 and 58: The variational equations describin
Page 59 and 60: (iii) Denote byη1i the i-th row of
Page 61 and 62: Then let H(Z, X, t) := X ′ − BX
Page 63 and 64: Proof. First, we work near q, then|
Page 65 and 66: This result can be written more com
Page 67 and 68: Proof. (of Theorem 2.4.2) The argum
Page 69 and 70: ˆZ is still exponentially small. P
Page 71 and 72: in the FitzHugh-Nagumo equation ǫ
Page 73 and 74: center-unstable manifold W cu (0, 0
Page 75 and 76:
where the second “fast jump” oc
Page 77 and 78:
CHAPTER 3 PAPER I: “HOMOCLINIC OR
Page 79 and 80:
[39, 40, 41, 42]). It states that f
Page 81 and 82:
papers. Geometric singular perturba
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s 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0
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One can view this as a projection o
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for x1. We find that there are thre
Page 89 and 90:
We compute the functionsγl andγr
Page 91 and 92:
3.3.3 Two Slow Variables, One Fast
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Also recall that the y-nullcline pa
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is negative for all values h∈(0,
Page 97 and 98:
3.4 The Full System 3.4.1 Hopf Bifu
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check whether this observation of a
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indistinguishable numerically from
Page 103 and 104:
we should view this curve as an app
Page 105 and 106:
s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 sin
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waves” (see e.g. [63, 15, 69]). 3
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the singular limits but it cannot b
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Then (3.24) transforms to a three t
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with x∈R m the vector of fast var
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ons that was used by David Terman i
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0.15 0.1 0.05 0 −0.05 0.15 0.1 0.
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from the input data. (If b−a is a
Page 121 and 122:
This explicit solution provides a b
Page 123 and 124:
4.4.2 Traveling Waves of the FitzHu
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computing the homoclinic orbit, eve
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y 0.15 0.1 0.05 0 −0.05 ε=0.001,
Page 129 and 130:
eturns directly to q. Figure 4.5(b)
Page 131 and 132:
are complicated [107]: we expect th
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t∈[0, 1]: z ′ = T F(z) H0(z(0))
Page 135 and 136:
CHAPTER 5 PAPER III: “HOMOCLINIC
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s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 H
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For a point p∈C0 we say that C0 i
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conjugate pair of eigenvalues for t
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Proof. (Sketch) The Lyapunov coeffi
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Table 5.1: Euclidean distance in (p
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gency of W s (q) with E u (Cl,ǫ).
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y x2 W s (q) C0 Cl,ǫ x1 W s (Cl,ǫ
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periodic orbit and q are restricted
Page 153 and 154:
yΣ1=ψ(Σ1) andΣ2=ψ(Σ2); see Fi
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eigenvalues, Shilnikov [102] proved
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s 1.39 1.385 1.38 1.375 1.37 1.365
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Figure 5.9(a)-(b). It was obtained
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was carried out with the stiff solv
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6.2 Introduction Our framework in t
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are called fold points. We can use
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curve C= Cl∪{(0, 0)}∪ Cr where
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explicitly. Note that currently no
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A slight modification of the formul
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6.5 Relating l1 and K Krupa and Szm
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nates. The computer algebra system
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atλ=λH= 0 forǫ= 0.05 is l MC 1
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imum and maximum of the cubic. The
Page 181 and 182:
6.8 Additions It is important to no
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−7.85 −7.95 −8.05 λ x −7.9
Page 185 and 186:
BIBLIOGRAPHY [1] V.I. Arnold. Encyc
Page 187 and 188:
[26] M. Desroches, B. Krauskopf, an
Page 189 and 190:
icated to Floris Takens. Eds.: Henk
Page 191 and 192:
[74] T.J. Kaper and C.K.R.T. Jones.
Page 193:
[98] H.G. Rotstein, M. Wechselberge
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